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Advanced Factorization

Definition

Factorization is the decomposition of an expression into a product of its factors.

The following are common factorizations.

  1. For any positive integer \(n\), \[a^n-b^n = (a-b)(a^{n-1} + a^{n-2} b + \ldots + ab^{n-2} + b^{n-1} ).\] In particular, for \( n=2\), we have \( a^2-b^2=(a-b)(a+b)\).

  2. For \( n\) an odd positive integer, \[ a^n+b^n = (a+b)(a^{n-1} - a^{n-2} b + \ldots - ab^{n-2} + b^{n-1} ).\]

  3. \( a^2 \pm 2ab + b^2 = (a\pm b)^2\)

  4. \( x^3 + y^3 + z^3 - 3 xyz = (x+y+z) (x^2+y^2+z^2-xy-yz-zx)\)

  5. \( (ax+by)^2 + (ay-bx)^2 = (a^2+b^2)(x^2+y^2)\). \( (ax-by)^2 - (ay-bx)^2 = (a^2-b^2)(x^2-y^2)\).

  6. \( x^2 y + y^2 z + z^2 x + x^2 z + y^2 x + z^2 y +2xyz= (x+y)(y+z)(z+x)\).

Factorization often transforms an expression into a form that is more easily manipulated algebraically, that has easily recognizable solutions, and that gives rise to clearly defined relationships.

Worked Examples

1. Find all ordered pairs of positive integer solutions \( (x,y)\) such that \(2^x+ 1 = y^2\).

Solution: We have \(2^x = y^2-1 = (y-1)(y+1)\). Since the factors \((y-1)\) and \((y+1)\) on the right hand side are integers whose product is a power of 2, both \((y-1)\) and \((y+1)\) must be powers of 2. Furthermore, their difference is

\[ (y+1)-(y-1)=2,\]

implying the factors must be \(y+1 = 4\) and \(y-1 = 2\). This gives \( y=3\), and thus \(x=3\). Therefore, \((3, 3)\) is the only solution.

2. Factorize the polynomial

\[f(a, b, c) = ab(a^2-b^2) + bc(b^2-c^2) + ca(c^2-a^2).\]

Solution: Observe that if \( a=b\), then \(f(a, a, c) =0\); if \(b=c\), then \(f(a, b, b)=0\); and if \( c=a\), then \( f(c,b,c)=0\). By the Remainder-Factor Theorem, \( (a-b), (b-c),\) and \( (c-a)\) are factors of \( f(a,b,c)\). This allows us to factorize

\[f(a,b,c) = -(a-b)(b-c)(c-a)(a+b+c).\]

Note by Calvin Lin
3 years, 4 months ago

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For the worked example - 1 : there are 2 solutions : (3,3) and (3,-3)

(y+1)-(y-1) = 2

implies y + 1 = -2 or y + 1 = 4 and y -1 = -4 or y-1 = 2

Thus, the two solutions you have are (3,3) and (3,-3) Kushashwa Shrimali · 3 years, 1 month ago

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@Kushashwa Shrimali Thanks. I added in "positive integers". Calvin Lin Staff · 2 years, 1 month ago

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I did not understand why should (y+1) - (y-1) = 2 ?
Can anyone explain ? Raven Herd · 2 years, 1 month ago

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@Raven Herd What do you not understand?

What do you think \( (y+1) - (y-1) \) is equal to ? Calvin Lin Staff · 2 years, 1 month ago

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@Calvin Lin I did not see it correctly . My fault ! Sir, I have a problem. I want to learn Number theory as is organized here on Brilliant but I don't follow anything beginning from modular inverses.I have tried the wikis but I still don't follow .Please suggest something. Raven Herd · 2 years, 1 month ago

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Thanks for this. Anuj Shikarkhane · 3 years, 1 month ago

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How to think that both difference is 2. What is the main moto behind thinking such that???? Prashant Kochar · 3 years, 1 month ago

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How is their (y+1)-(y-1) difference 2? Ishan Saxena · 3 years, 1 month ago

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@Ishan Saxena \[(y+1) - (y-1)\] \[= y +1 - y + 1\] \[= 1+1= \boxed{2} \] Ankita Mehra · 3 years, 1 month ago

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@Ishan Saxena (Y+1)-(y-1) = y+1-y+1= 2 Abhishek Gupta · 3 years, 1 month ago

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@Ishan Saxena 2 Sandeep Patel · 3 years, 1 month ago

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